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ACMS Abstracts: Spring 2016

Stefan Llewellyn Smith (UCSD)

Hollow vortices

Hollow vortices are vortices whose interior is at rest. They posses vortex sheets on their boundaries and can be viewed as a desingularization of point vortices. After giving a history of point vortices, we obtain exact solutions for hollow vortices in linear and nonlinear strain and examine the properties of streets of hollow vortices. The former can be viewed as a canonical example of a hollow vortex in an arbitrary flow, and its stability properties depend on a single non-dimensional parameter. In the latter case, we reexamine the hollow vortex street of Baker, Saffman and Sheffield and examine its stability to arbitrary disturbances, and then investigate the double hollow vortex street. Implications and extensions of this work are discussed.

Tom Solomon (Bucknell)

Experimental studies of reaction front barriers in laminar flows

We present studies of the effects of vortex-dominated fluid flows on the motion of reaction fronts produced by the excitable Belousov-Zhabotinsky reaction. The results of these experiments have applications for advection-reaction-diffusion dynamics in a wide range of systems including microfluidic chemical reactors, cellular-scale processes in biological systems, and blooms of phytoplankton in the oceans. To predict the behavior of reaction fronts, we adapt tools used to describe passive mixing.In particular, the concept of an invariant manifold is extended to account for reactive burning. Burning invariant manifolds (BIMs) are predicted as one-way barriers that locally block the motion of reaction fronts. These ideas are tested and illustrated experimentally in a chain of alternating vortices, a spatially-random flow, vortex flows with imposed winds, and a three-dimensional, nested vortex flow. We also discuss the applicability of BIM theory to the motion of bacteria in fluid flows.

Daniele Cappelletti (KU)

Deterministic and stochastic reaction networks

Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development.

Lihui Chai (UCSB)

Semiclassical limit of the Schrödinger-Poisson-Landau-Lifshitz-Gilbert system

The Schrödinger-Poisson-Landau-Lifshitz-Gilbert (SPLLG) system is an effective microscopic model that describes the coupling between conduction electron spins and the magnetization in ferromagnetic materials. This system has been used in connection to the study of spin transfer and magnetization reversal in ferromagnetic materials. In this paper, we rigorously derive the Vlasov-Poisson-Landau-Lifshitz-Glibert system as the semiclassical limit of SPLLG. The major difficulties come from the presence of the spin-magnetization coupling and the discontinuities of the magnetization at the boundary of the material. To overcome these difficulties, we first take the semiclassical limit (vanishing Planck constant) of a smoothed SPLLG system, and then the limit of vanishing smoothing parameter. As a byproduct, we prove the local existence and uniqueness of classical solutions to the smoothed SPLLG system.

Alejandro Roldan-Alzate (UW)

Non–invasive patient-specific cardiovascular fluid dynamics

Comprehensive characterization and quantification of blood flow is essential for understanding the function of the cardiovascular system under normal and diseased conditions. This provides important information not only for the diagnosis and treatment planning of different cardiovascular diseases but also for the design of cardiovascular devices. However, the anatomical complexity and multidirectional nature of physiological and pathological hemodynamics makes non-invasive characterization and quantification of blood flow difficult and challenging. Doppler ultrasound, a standard imaging technique, is limited to providing information on large vessels and calculating instantaneous average flow within the cardiac cycle. Magnetic resonance imaging (MRI) is increasingly being used for fluid dynamics analyses of cardiovascular diseases, including pulmonary arterial hypertension, portal hypertension and congenital heart diseases. Although two-dimensional (2D) phase contrast (PC) magnetic resonance imaging (MRI) measures velocity across a plane, it is still limited in its ability to fully characterize these complex flow systems. Four- dimensional (4D) flow MRI obtains velocity measurements in three dimensions throughout the entire cardiac cycle. Several attempts have been made to non-invasively characterize the blood flow dynamics of different cardiovascular diseases using the combination of medical imaging and computational fluid dynamics modeling (CFD). Idealized geometries, as well as patient-specific anatomies, have been used for computational simulations, which have improved the understanding of the fluid dynamics phenomena in different vascular territories. While CFD modeling can provide powerful insights and the potential for simulating different physiological and pathological conditions in the cardiovascular system, it is currently not reliable for use in clinical care. Based on different studies, additional work is needed to verify the accuracy of current CFD approaches or identify and address current shortcomings. The overall purpose of this research is to develop, implement and validate non-invasive flow analysis methodologies to assess cardiovascular flow dynamics, using a combination of 4D flow MRI, numerical simulations and patient-specific physical models. In this seminar, multidisciplinary work will be presented first, where different cardiovascular pathologies have been studied, such as congenital heart disease and portal hypertension using in vivo, in vitro and computational models. Second, some advances will be presented and a future outlook into the valuable contribution of engineering in the medical imaging and diagnostic technology will be provided.